If I had to point to a single anime that really began pushing the music used for series to the forefront, it would be the Bubblegum Crisis OVAs of the late 1980's. Before that series came about, much of the OP and ED music you find in anime has more in common with traditional Japanese music (a.k.a. enka) than with anything that was actually popular at the time. Not to mention it's largely boring to listen to on its own. BGC's music really opened the door in more ways than one.

Really, though, if you look back on anime music as a whole, things didn't start getting better (nevermind memorable) until the mid-1990's. Many of the series we hold now as classics from that decade are the ones that provided us with the most memorable OPs, EDs and insert songs that we today might consider among the "greatest of all time". Consequently, this is why originally I said I understood why people might want to put an age minimum on songs that get nominated as being the greatest. These are series and songs that got many people into anime in the first place, after all. It's natural there would be some amount of leaning towards them.

However, take a moment and consider how far that would restrict things here. If we're saying a song needs to be at least 10 years old, then we're talking about anime that aired no more recently than 1999. Taking as a given, as I said earlier, that anime music prior to the early 90's was largely forgettable, you're talking about restricting our music selection for the greatest anime songs of all time to not much more than a single decade's worth of series to select from. That's not only unfair to the many memorable songs and series of this decade, it's far too narrow-minded.

Age has little, if anything, to do with the quality of a given song. Like janfive said, there are such things as instant classics. Good, memorable music should be recognized on its merits, not something arbitrary such as how long it's been around.